Decomposition of some Witten-Reshetikhin-Turaev Representations into Irreducible Factors

We decompose into irreducible factors the ${\rm SU}(2)$ Witten-Reshetikhin-Turaev representations of the mapping class group of a genus $2$ surface when the level is $p=4r$ and $p=2r^2$ with $r$ an odd prime and when $p=2r_1r_2$ with $r_1$, $r_2$ two distinct odd primes. Some partial generalizations in higher genus are also presented.


Introduction
Witten gave in [Wit89] convincing arguments for the existence of Topological Field Theories, as defined in [Ati88,Wit88], giving a three dimensional interpretation of the Jones polynomial when the gauge group is SU(2). Each of these TQFTs gives a family of projective finite dimensional representations of the mapping class group Mod(Σ g ) of a genus g closed oriented surface Σ g . Reshetikhin and Turaev gave a rigorous construction of these TQFTs [RT91] using representations of quantum groups. In this paper we will follow the skein theoretical construction of [Lic91,BHMV95] to define these representations.
We can lift these projective representations to linear representations of some central extension Mod(Σ g ) of Mod(Σ g ) noted: ρ p,g : Mod(Σ g ) → GL(V p,g ).
Here p = 2(k + 2) ≥ 3 is an even integer indexing the representations and V p,g is a finite dimensional complex vector space. These representations are equipped with an invariant scalar product , p,g with respect to which they are unitary.
The goal of this paper is to decompose some of these representations into irreducible factors. Only few results are known concerning their decomposition. In [BHMV95], an explicit proper submodule of V p,g is given whenever 4 divides p. In [Rob01] it is shown that V p,g is irreducible when p 2 is an odd prime. Robert's proof extends word-by-word to show that the modules V 18,g are also irreducible. In [AF10] the authors showed that for p = 24, 36, 60 then

Skein construction of the Reshetikhin-Turaev representations
Following [BHMV95], we will briefly define the representations ρ p,g and fix some notations.

The spaces V p,g
Given an even integer p ≥ 6, we denote by A ∈ C an arbitrary primitive 2p − th roots of unity. Using the Kauffman skein relation of Figure 1, we associate to any framed link L ⊂ S 3 an invariant L p ∈ C. Choose g ≥ 1 and denote by C g the set of isotopy classes of framed links (including the empty link) in an oriented genus g handlebody H g . We fix a genus g Heegaard splitting of the sphere, i.e. an element S ∈ Mod(Σ g ) and two handlebodies so that : H 1 g S:∂H 1 g →∂H 2 g H 2 g S 3 Take L 1 , L 2 ∈ C g and embed L 1 in H 1 g and L 2 in H 2 g . The above gluing defines a link L 1 S L 2 ⊂ S 3 . We call Hopf pairing the bilinear form: Eventually we define the spaces V p,g as the quotients: The vector spaces V p,g are finite dimensional ( [BHMV95]) and we can find explicit basis as follows. Let g ≥ 2, choose a trivalent graph Γ ⊂ H g so that H g retracts on Γ by deformation. If g = 1, Γ represents the circle S 1 × {0} ⊂ S 1 × D 2 H 1 . We denote by E(Γ) the set of its edges.
In [Jon83,Wen87] the authors defined some idempotents { f 0 , . . . , f p−4 2 } of the Temperley-Lieb algebra with coefficient in Q(A) called Jones-Wenzl idempotents. To σ a p-admissible coloring of Γ we associate a vector u σ ∈ V p,g as follows. We replace each edge e ∈ E(Γ) by the Jones-Wenzl idempotent f σ(e) . If (e 1 , e 2 , e 3 ) are three edges adjacent to a vertex of Γ, we connect the idempotents using the link T σ(e 1 ),σ(e 2 ),σ(e 3 ) defined in Figure 2. The Theorem 4.11 of [BHMV95] asserts that the elements u σ , for σ a padmissible coloring of Γ, form a basis of V p,g . Moreover there exists a nondegenerate bilinear form , p,g on V p,g invariant under the action of Mod(Σ g ), for which the vectors u σ are pairwise orthogonal.
The basis u σ depends on the choice of the trivalent graph. We can transform a trivalent graph into one another by a sequence of Whitehead moves. Suppose that Γ 1 and Γ 2 are two trivalent graphs of genus g ≥ 2, which only differ by a single Whitehead move, inside a ball B 3 , as drawn in Figure 3. Fix a p-admissible coloring of the graphs outside B 3 and denote by σ(i) (resp. σ( j) ) the vector associated to the coloration of Γ 1 (resp of Γ 2 ) with the edge i colored by σ(i) (resp with the edge j colored by σ( j)).
Then the vectors i belong to the subspace spanned by the vectors j a b c d i j b a c d Figure 3: The two graphs Γ 1 on the left and Γ 2 on the right differ by a local Whitehead move.
and decompose using the so-called 'fusion rules' formula: where the sum runs through p-admissible colorings and the coefficient a b j c d i only depends on the colors of the edges a, b, c, d, i and j and is called recoupling coefficient or 6 j-symbol in literature. We refer to [MV94] for a proof and an explicit computation of these coefficients. Numerical experiments suggest that they are never null.

The Reshetikhin-Turaev representations
We fix an orientation preserving homeomorphism Choose a class φ ∈ Mod(Σ g ) associated to a homeomorphism which extends to H g through α. Then φ acts on C g and preserves the kernel of the Hopf pairing so acts on V p,g by passing to the quotient. Denote byρ p,g (φ) ∈ GL(V p,g ) the resulting operator. Now choose φ ∈ Mod(Σ g ) so that the corresponding homeomorphisms extend to H g through α•S. This extension also defines, by quotient, an operator on V p,g . We denote byρ p,g (φ) the dual of this operator for the Hopf pairing.
The elements of Mod(Σ g ) which extend to H g either through α or through α • S, generate the whole group Mod(Σ g ). It is a non trivial fact that the associated operatorsρ p,g (φ) generate a projective representation: We consider a central extension Mod(Σ g ) of Mod(Σ g ) that lifts the above projective representations to linear ones (see [MR95,GM09]): These are the so-called Reshetikhin-Turaev representations. Now to each edge e ∈ E(Γ), choose a disc D e , properly embedded in H g , that intersects Γ transversely once in e. Note that the set of boundary curves γ e := ∂D e ⊂ ∂H g α − → Σ g forms a pants decomposition of Σ g . A classical property of the Jones-Wenzl idempotents asserts that, if T e ∈ Mod(Σ g ) denotes the Dehn twist along γ e , then: We fix the lift of T e in Mod(Σ g ), still denoted T e , so that ρ p,g (T e ) · u σ = µ σ(e) u σ .
We also fix the lift S ∈ Mod(Σ g ) so that the matrix of ρ p,g (S) is the matrix of the Hopf pairing (·, ·) H p,g multiplied by an element η ∈ C which verifies |η| = |A 2 −A −2 | √ p . We refer to [BHMV95], where η represents the quantum invariant of S 3 , for a detailed discussion on η.
Since S and the {T e } e∈E(Γ) generate Mod(Σ g ) for some trivalent graphs, we have an explicit description of ρ p,g .

Cyclicity of the vacuum vector
Denote by A p,g the subalgebra of End(V p,g ) generated by the operators ρ p,g (φ) for φ ∈ Mod(Σ g ). The key ingredient to prove Theorem 1.1 is to show that the vacuum vector v 0 ∈ V p,g , associated to the class of the empty link, is cyclic, i.e. that A p,g · v 0 = V p,g .

The genus one case
In [Kor13] we gave an explicit decomposition of the Weil representations into irreducible factors. An easy generalization of the arguments of the proof of Lemma 3 of [FK06] leads to an explicit isomorphism of SL 2 (Z)-modules between V p,1 and the odd submodule of the Weil representation at level p. Proving that v 0 ∈ V p,1 is cyclic reduces to show that its projection on each irreducible submodule of V p,1 is not null.
Denote by u 0 , . . . , u p−4 2 the basis of V p,1 where u i is the class of the closure of the i − th Jones-Wenzl idempotent along a longitude in H 1 . Also denote by e i , i ∈ Z/pZ the basis of the Weil SL 2 (Z)-module U p at level p as described in [Kor13].
In this basis, the Weil projective representations in genus one are defined by the matrices: Here the level is an integer p ≥ 2 not necessary even. When p is even, we take A to be a primitive 2p − th roots of unity. When p is odd, A is a primitive p − th roots of unity.
Lemma 3.1. Let p = 2r ≥ 6 be an even integer. Then the following map: Proof. We compute the matrix elements: The decomposition into irreducible submodules of U p is described by the following: Kor13]). We have the following decompositions where denotes an isomorphism of SL 2 (Z)-modules: 1. If a and b are coprime, then U ab U a ⊗ U b .
2. If r is prime and n ≥ 1, then U r n+2 U r n ⊕ W r n+2 where W r n+2 denotes another module.
3. If r is an odd prime, then U r 2 1 ⊕ W r 2 where 1 is the trivial representation.
4. The modules U p for r > 2 and W r n split into two submodules: or W − r n and have pairwise coprime levels, are irreducible.
We can now prove: Proposition 3.3. Let p ≥ 6 be an even integer. Then the vacuum vector v 0 ∈ V p,1 is cyclic if and only if one of the following three cases holds: • p = 2r 1 . . . r k with r i distinct odd primes.
Proof. We will use Proposition 3.2 and the explicit isomorphisms given in the main Theorem of [Kor13] to study whether the vector has non trivial projection on each submodule of U − p or not. Given two integers x and n, we will denote by [x] n ∈ Z/nZ the class of x modulo n.
2 . First, when p = 2r 2 , with r prime, the module U − p is irreducible so the vector is cyclic.
When p = 4r, with r an odd prime, the module decomposes into two irreducible submodules: Where the first term lies in U − 4 ⊗ U + r and the second in When p = 2r 1 . . . r k , with r i distinct odd primes, we have the following decomposition: Let us fix ǫ and denote: where we used the notation e ± i := e i ± e −i . By using the facts that e i , e ǫ i = 1 and So the projection of v on each irreducible submodule X ǫ is not null.
Now suppose that p = 2r n 1 1 . . . r n k k with k ≥ 2, r i distinct primes and n 1 ≥ 2. Since r 1 does not divide x, the vector v has a null projection on the submodule: Next if p = 2r n , with r an odd prime and n ≥ 2, the projection of v on

Cyclicity in higher genus
The goal of this subsection is to prove the following: Proposition 3.4. When g ≥ 2, the vacuum vector v 0 ∈ V p,g is cyclic in the following cases: 1. When p = 4r with r an odd prime and if g = 2 or if p is generic and g < r − 2.
2. When p = 2r 2 with r an odd prime and g = 2 or if p = 50 and g = 3.
Fix a trivalent graph Γ ⊂ H g as in section 2. Two p-admissible colorings σ 1 , σ 2 of Γ will be said equivalent if: We denote by col p (Γ) the set of equivalence classes of colorings for this relation. To [σ] ∈ col p (Γ), we associate the subspace: Proof. The matrices ρ p,g (T e ), for e ∈ E(Γ), generate a commutative subalgebra of A p,g . The set col p (Γ) indexes its characters and the spaces W [σ] are the associated common eigenspaces of the ρ p,g (T e ). The orthogonal projector on X must commute with the ρ p,g (T e ) and thus preserves the subspaces W [σ] .
The strategy to prove Proposition 3.4 is to apply Lemma 3.5 to ⊥ the orthogonal (for the invariant form) of the cyclic space generated by the vacuum vector.
1. We call Γ g a fly eyes graph of genus g if it is a trivalent graph obtained by the following inductive method: • Γ 2 is the Theta graph .
• A graph Γ g+1 is obtained from a Γ g by choosing arbitrary a vertex and inserting a triangle as drawn on the left-hand side of Figure 4.
The right-hand side gives an example of a genus 8 fly eyes graph.
(a) (b) Figure 4: On the left: the operation transforming a fly eyes graph of genus g into a one of genus g + 1. On the right: an example of genus 8 fly eye graph.
2. The genus 3 fly eyes graph is unique and is called the tetrahedron graph. We say that a level p ≥ 3 is generic if for any coloring σ of Γ 3 , we have: The complex numbers (u σ , v 0 ) H p,3 are called tetrahedron coefficients in literature and are related to the 6 j-symbols defined in the previous section. In particular it is equivalent to say that the 6 j-symbols or the tetrahedron coefficients are not null for a level p. It follows from fusion-rules (equation (1)) that if p is generic, then for any g ≥ 3, for any fly eyes graph Γ g and for any p-admissible coloring σ of Γ g , we have: Fix g ≥ 2 and embed a fly eyes graph Γ g in S 3 . Denote by H g the embedded handlebody where V(Γ g ) denotes a tubular neighborhood of Γ g . For each edge e ∈ E(Γ g ), fix a curve γ e ⊂ H g which bounds a disc intersecting Γ g only once along e.
We construct a map: as follows.
To f : E(Γ g ) → N we associate the class in V p,g of the link made of f (e) parallel copies of γ e for each edge e ∈ E(Γ g ).
When g = 2, we will note w a,b,c ∈ V p,2 the class of the link made of a parallel copies of γ 1 , b copies of γ 2 and c of γ 3 .
The Figure 5 shows the curves γ e when g = 2 and g = 3. Lemma 3.7. If p = 4r, with r an odd prime, or if p = 2r 1 r 2 , with r 1 , r 2 two distinct odd primes, then: w a,b,c ∈ A p,2 · v 0 , for any a, b, c ∈ {0, 1} Proof. Choose a longitude L and a meridian M of Σ 1 . The space A p,1 · v 0 = V p,1 is generated by juxtaposition of properly embedded parallel copies of L and M in H 1 , colored by the element ω ∈ V p,1 as defined in [BHMV92]. By embedding the skein elements L(ω) and M(ω) in H 2 H 1 #H 1 in both handles, we see that the vectors w i,0, j belong to A p,2 · v 0 for arbitrary i and j. So do the vectors w i, j,0 by action of the mapping class group.
Using fusion rules (see [MV94]), we have that: where T e is (a lift of) the Dehn twist around the middle edge of the Theta graph (labeled 0). Since both vectors belong to V p,2 · v 0 and since the recoupling coefficients 2 2 k 2 2 0 and 2 2 k 2 2 2 are not zero and µ 2 1, we know that the 3-dimensional space generated by 2 0 2 , 2 2 2 and 2 4 2 is included in A p,2 · v 0 . So does the vector 2 2 2 .
Lemma 3.8. When p = 2r 2 , with r an odd prime, then Moreover, if σ is a p-admissible coloring of Γ = such that: σ(e) −1 (mod r), for all e ∈ E(Γ) Proof. Note first that i, j ∈ 0, . . . p−4 2 are such that: if and only if i ≡ j ≡ −1 (mod r) and i and j have same parity (and are distinct). Thus when σ satisfies the condition of the Lemma, the subspace W [σ] is onedimensional. The Lemma 3.5 implies that this subspace is either in A p,2 · v 0 , or in its orthogonal. Now note that the Hopf pairing (u σ , v 0 ) H p,2 is not zero for it is equal to a 3 j-symbol. This prove the second part of the Lemma.
In particular, we just proved that: We finish the proof by noticing that the vector w a,b,c belongs to this space whenever we have: Lemma 3.9. The vector w f belongs to A p,g · v 0 for f ∈ {0, 1} E(Γ g ) when p is generic and: • p = 4r with r an odd prime such that g ≤ r − 2.
Proof. We proceed like in the proof of Lemma 3.8: first we note that if f ∈ {0, 1} E(Γ g ) then: Then we note that if σ is such that 0 ≤ σ(e) ≤ g for all e ∈ Γ g , then The fact that these W [σ] are one-dimensional is deduced from the following two facts: 1. When p = 4r, and i, j ∈ {0, . . .

Proof of Proposition 3.4. Fix a fly eyes graph Γ, a class [σ] ∈ col p (Γ), and choose a vector
⊥ By Lemma 3.5, we must show that v = 0 to conclude. We will find dim W [σ] independent equations verified by the coefficients α σ ′ .
Using Lemmas 3.7, 3.8 and 3.9, we know that By definition of v, we have that: Since the complex numbers (u σ ′ , v 0 ) H p,g are non null when p is generic, it is enough to show that the matrix: f ∈F has independent lines to conclude the proof.
We now define an invertible square matrixM such that M is obtained from M by removing some lines.
When i ∈ 0, . . . , is clearly invertible and M is obtained fromM by removing the lines corresponding to non p-admissible colorings of Γ.

Decomposition into irreducible factors
In this section, we will prove the Theorems 1.1 and 1.3. Denote by (A p,g ) ′ the commutant of the algebra A p,g , i.e. the subspace of End(V p,g ) of operators commuting with all the ρ p,g (φ) for φ ∈ Mod(Σ g ). The dimension of (A p,g ) ′ is equal to the number of irreducible submodules of V p,g . We thus have to show that dim (A p,g ) ′ is one if p = 2r 2 and p = 2r 1 r 2 and is two when p = 4r with the additional assumptions of the two Theorems.
Consider the following linear map: The cyclicity of v 0 (Proposition 3.4) implies that f is injective. Moreover if φ ∈ Mod(Σ g ) is the lift of a homeomorphism of Σ g that extends to H g through α : Σ g → ∂H g , then: Denote by Mod(H g ) ⊂ Mod(Σ g ) the subgroup generated by these φ. By definition, we have: In particular, for any trivalent graph Γ, we have Range( f ) ⊂ W [0] (Γ) where [0] is the class of the coloring sending every edges of Γ to 0. As an immediate consequence, we get the: Proof of Theorems 1.1 and 1.3 when p = 2r 2 . When p = 2r 2 , with r an odd prime, then W [0] is one-dimensional, generated by v 0 . Thus Range( f ) = {v 0 } and (A 2r 2 ,g ) ′ = {1}. The Schur Lemma implies that the module V 2r 2 ,g is irreducible.
Remark. When p = 50, we remark that the numbers µ 0 , µ 1 , . . . , µ 23 are pairwise distinct. The proof of Roberts [Rob01] applies word-by-word in this case to show that V 50,g is irreducible. Indeed the fact that the µ i are distinct implies that the null vector v 0 ∈ V 50,1 is cyclic for the action of the group generated by the Dehn twist along the longitude of H 1 . This easily implies that v 0 ∈ V 50,g is cyclic for the action of Mod(Σ g ) for arbitrary g ≥ 1 and we conclude as above by noticing that W [0] is one dimensional generated by v 0 .

The case where p = 4r
Let p ≥ 3 be such that p ≡ 4 (mod 8). Consider a link L ⊂ Σ g × { 1 2 } inside the cylinder Σ g × [0, 1] and color L by p parallel copies of ω or, equivalently, by the p−4 2 − th Jones-Wenzl idempotent. The gluing of the above cobordism on H g induces an operator acting on V p,g . In [BHMV95] it is shown that this operator only depends the homology class of L in H 1 Σ g , Z/2Z and we get this way an injective morphism of algebras: Its action on v 0 gives the space We denote by P the projector of V p,g on the subspace of vectors fixed by the operators of i(C H 1 Σ g , Z/2Z ). Clearly P ∈ (A p,g ) ′ .
Note x i , y i ∈ H 1 Σ g , Z/2Z the meridian and longitude around the i − th hole and note: The Θ i 's are symmetries which pairwise commute and The symmetric group σ g acts by permutation on the generators of C[Θ 1 , . . . , Θ g ].
We note W g ⊂ i C H 1 Σ g , Z/2Z the subalgebra of C[Θ 1 , . . . , Θ g ] of elements fixed by σ g .
Finally we denote by I ⊂ i C H 1 Σ g , Z/2Z the ideal generated by the elements (x i − 1) for 1 ≤ i ≤ g. We have: Lemma 4.1. Consider the action of Sp 2g, Z/2Z on i C H 1 Σ g , Z/2Z . Then: 1. The vectors fixed by this action are the ones of Span(1, P).
2. For every w ∈ W g and φ ∈ Sp 2g, Z/2Z we have: Proof. The first point follows from the fact that the action of Sp 2g, Z/2Z on H 1 Σ g , Z/2Z has two orbits: the singleton containing the neutral element and the set containing the other elements. Indeed by taking an appropriate Z/2Zbasis of H 1 Σ g , Z/2Z , this action is described by the usual Birman generators of Sp(2g, Z) ( [Bir71]) passed to the quotient in Sp 2g, Z/2Z , that is the 2g × 2g matrices: 1 g B 0 g 1 g and 0 g 1 g 1 g 0 g where A ∈ GL(g, Z/2Z) and B is symmetric. We just have to remark that the commutant of the algebra generated by these matrices consists of the scalar matrices to conclude.
To prove the second point, denote by X i , Y i , Z i, j for 1 ≤ i, j ≤ g the class in H 1 Σ g , Z/2Z of the Dehn twists of Figure 6 generating H 1 Σ g , Z/2Z . First note that the operators Θ i are invariant under the action of the X i and Y i and that the element of the algebra W g are invariant under permutation of the handles. We are reduced to show that for w ∈ W g , we have Z 1,2 · w − w ∈ I. Figure 6: Some Dehn twists generating Sp 2g, Z/2Z when g = 3 by passing to the quotient.
First note that Z 1,2 · Θ i = Θ i when i {1, 2}. Then we compute: The case p = 4r of Theorems 1.1 and 1.3 are easily deduced from the: Proposition 4.2. If p ≡ 4 (mod 8) and v 0 ∈ V p,g is cyclic, then Since Θ·v 0 lies in W [0] and is invariant under permutation of the handles, there exists an element w ∈ W g such that w · v 0 = Θ · v 0 . Now if φ ∈ Mod(Σ g ), then: where we used the second point of Lemma 4.1 in the last equality. Using the cyclicity of v 0 we get that Θ = w ∈ W g . We conclude using the first point of Lemma 4.1.
We begin by stating a technical Lemma which proof will be the subject of the next subsection: Lemma 4.3. If (x, x, x) is p-admissible, then we have the following: Lemma 4.4. Let p ≥ 3 be such that (x, x, x) is p-admissible. Let Γ 1 , Γ 2 be two trivalent graphs which only differ by a single Whitehead move inside a ball B 3 as drawn in Figure  3. Then: Proof. Let σ 1 , σ 2 be two p-admissible colorings of Γ 1 , with colors 0 or x, such that: σ 1 (e) = σ 2 (e), ∀e ∈ E(Γ 1 ) − {i} Suppose there exists (α, β) ∈ C 2 so that: We must show that α = β = 0 to conclude. Using the fusion rule equation (1) of section 2.1, we get: where 2 and 4 represent the vectors associated to colorations of Γ 2 by the same colors that σ 1 , σ 2 outside the ball B 3 and with the edge j colored respectively by 2 and 4. The vector v ′ is orthogonal to the two previous ones. Now since v ∈ W [0] (Γ 2 ), we have the following system: We conclude using Lemma 4.3 If i ∈ {1, . . . , g}, we note b i ∈ V p,g the vector representing a single ribbon colored by x around the i − th hole.
Lemma 4.5. If G g represents the set of all trivalent graph of genus g, then: Proof. Let σ be a coloring of the graph of Figure 7 such that: 1. σ(e) ∈ {0, x}, for all e ∈ E(Γ), We can suppose that for every i < k < j, then σ(a k )σ(b k ) = 0. Using Lemma 4.4 with a = a i , b = a j , c = b i and d = b j , we have that the projection of u σ on Γ∈G g W [0] (Γ) is null.
We conclude by noticing that if σ is a coloring of Γ, with colors in {0, x}, that does not satisfies 2, then u σ = b i for some i ∈ {1, . . . , g} or u σ = v 0 .
Lemma 4.6. There exists an element a ∈ A 2r 1 r 2 ,1 so that: Proof. It is enough to show that there exists a symmetry ψ ∈ (A 2r 1 r 2 ,1 ) ′ so that: ψ · u 0 = u x and ψ · u x = u 0 Indeed, the cyclicity of u 0 (Proposition 3.3) implies the existence of a ∈ A 2r 1 r 2 ,1 so that a · u 0 = u x If such a ψ does exist, we then have: The symmetry ψ is defined as follows: choose i ∈ {0, . . . , r 1 r 2 − 2}, then only one of the following two cases occurs: • Either there exists j ∈ {0, . . . , r 1 r 2 − 2} so that j ≡ i (mod 2r 1 ) j ≡ −i − 2 (mod r 2 ) and we put ψ(u i ) := +u j .
A straightforward computation shows that ψ commutes with ρ p,1 (T) and ρ p,1 (S) and either ψ or −ψ sends u 0 to u x .
The proof of the Theorems 1.1 and 1.3 when p = 2r 1 r 2 follows from the following: Proposition 4.7. Let r 1 , r 2 be two distinct odd primes, p = 2r 1 r 2 and g ≥ 2 be such that v 0 ∈ V p,g is cyclic. Then V p,g is irreducible.
Proof. Using Lemma 4.5 and the fact that the vectors of Range( f ) must be invariant under permutation of the handles, we have that: By contradiction, suppose there exists Θ ∈ (A p,g ) ′ so that: where a ⊗ 1 ⊗ . . . ⊗ 1 denotes the embedding of the element a ∈ A p,1 , seen as a linear combination of ω-colored link in Σ 1 ×[0, 1], in the first handle of Σ g ×[0, 1].

Proof of Lemma 4.3
In this subsection we put p = 2r 1 r 2 , with r 1 , r 2 two distinct odd primes. We suppose there exists x ∈ {1, . . . , r 1 r 2 − 2} so that (x, x, x) is p-admissible and so that x ≡ 0 (mod 2r 1 ) x ≡ −2 (mod r 2 ) We also choose A 1 and A 2 some primitive r 1 − th and r 2 − th roots of unity, so that A 2 = A 1 A 2 . In particular we have A 2x = A −2 2 . The goal of this subsection is to show that: So we just have the following possible cases: